The Likelihood

The data set $D$ is the disjunction of the set $D_0$ of results of the present experiments (chapter 5,) the set $D_{\Gamma}$ of published $\Gamma$ calibration data (figure 5.6,) and the set $D_S$ of published $S$ calibration data (figure 5.7.)

For each measured electron arrival rate $f_m^{(i, j)} \in{} D_0$, the expectation of $f_m^{(i, j)}$, with a given parameter vector $\mathbf{Q}$, is $f_p^{(i, j)}$ (equation 5.25,) calculated using parameter vector $\mathbf{Q}$, and the values of $E_b$ and $F$ pertaining to the data point in question. There are two known, significant sources of uncertainty in this prediction. The first results from the Poisson process [84] of discrete electron arrivals at the channeltron, and introduces into the measured arrival rate a standard deviation $\sqrt{\frac{f_p^{(i, j)}}{256\tau{}}}$, where $256\tau$ is the time interval, over which electrons are counted to obtain the arrival rate $f_m^{i, j}$. The second results from the standard deviation uncertainty $\Delta{}F$, in the incident beam current $F$, and introduces into the measured arrival rate a standard deviation $\frac{f_p^{(i, j)}\Delta{}F}{F}$. These are combined in quadrature to give an overall standard deviation $\sqrt{\frac{f_p^{(i, j)}}{256\tau{}}+\frac{f_p^{(i, j)2}\Delta{}F^2}{F^2}}$. The likelihood function is then taken to be

$$P(f_m^{(i, j)}\vert\mathbf{Q}) = G\left(f_m^{(i, j)}; f_p^{(... ...tau{}}+\frac{f_p^{(i, j)2}\Delta{}F^2}{F^2}}\right)\textrm{.}$$ (5.36)

For a given parameter vector, the measurements are assumed to be independent:

$$P(D_0\vert\mathbf{Q}) = \prod_{f_m^{(i, j)}\in{}D_0}P(f_m^{(i, j)}\vert\mathbf{Q})\textrm{.}$$ (5.37)

For each measured spin-averaged scattering probability $\Gamma_E \in{} D_{\Gamma}$, the expectation of $\Gamma_E$, with a given parameter vector $\mathbf{Q}$, is $\Gamma$ (equation 5.23,) calculated using the $\Gamma_k$ from $\mathbf{Q}$, and the value of $W$ relevant to $\Gamma_E$. The uncertainty is introduced by the quantization, in units of $q_{\Gamma} = 0.001$, of the author's readings from the published [6] graphs. This gives

$$P(\Gamma_E\vert\mathbf{Q}) = T\left(\Gamma_E; \Gamma{}-\frac{q_{\Gamma}}{2}, \Gamma{}+\frac{q_{\Gamma}}{2}\right)\textrm{.}$$ (5.38)

For a given parameter vector, the measurements are assumed to be independent:

$$P(D_{\Gamma}\vert\mathbf{Q}) = \prod_{\Gamma_E\in{}D_{\Gamma}}P(\Gamma_E\vert\mathbf{Q})\textrm{.}$$ (5.39)

For each measured Sherman function $S_E \in{} D_S$, the expectation of $S_E$, with a given parameter vector $\mathbf{Q}$, is $S$ (equation 5.24,) calculated using the $S_k$ from $\mathbf{Q}$, and the value of $W$ relevant to $S_E$. As well as the published [6] uncertainty $\Delta{}S_E$, uncertainty is introduced by the quantization, in units of $q_S = 0.025$, of the author's readings from the published [6] graphs: the standard deviation associated with this is $\frac{q_S}{\sqrt{12}}$. These are combined in quadrature, to give a standard deviation $\sqrt{\Delta{}S_E^2+\frac{q_S^2}{12}}$. The likelihood is then taken to be

$$P(S_E\vert\mathbf{Q}) = G\left(S_E; S, \sqrt{\Delta{}S_E^2+\frac{q_S^2}{12}}\right)\textrm{.}$$ (5.40)

For a given parameter vector, the measurements are assumed to be independent:

$$P(D_S\vert\mathbf{Q}) = \prod_{S_E\in{}D_S}P(S_E\vert\mathbf{Q})\textrm{.}$$ (5.41)

For a given parameter vector, the three data sets are assumed to be independent:

$$P(D\vert\mathbf{Q}) = P(D_0\vert\mathbf{Q})P(D_{\Gamma}\vert\mathbf{Q})P(D_S\vert\mathbf{Q})\textrm{.}$$ (5.42)